“Abraham, Martin, and John” by Dion.
Oh, I forgot “Strange Fruit,” by Billie Holiday. Also “Comfortably Numb,” by Pink Floyd.
The Needle and the Damage Done by Neil Young
Hollow by Pantera
Ooh, ooh, another one. Motherless Child makes me cry. 
You stole mine! Oh well, I’ll second this one.
That song is more than just depressing, it’s downright scary.
My suggestions are “Bleeker Street”, “Sparrow” “The Sounds of Silence” and “I Am A Rock” from Simon and Garfunkel
And the original by Nine Inch Nails
Tomorrow Wendy by Concrete Blonde
If you’re into this kind of thing Nightmare Launch by Leslie Fish
“Pool Shark” by Sublime
Now I’ve got the needle
and I can’t shake
but I can’t breathe
I take it away but I want more and more
One day I’m gonna lose the war
He died a year or two later of a heroin overdose.
“The Drunk Driver” by Red Sovine
“The Blind Man in the Bleachers” by Dave Geddes
"The Green Fields of France" as written and sung by Eric Bogle, although the Fureys’ recording is more common.
On the subject of Simon and Garfunkel, “The Dangling Conversation” is another one.
I always thought that “The Dangling Conversation” was about a failed marriage:
Is it not?
Aside from *Sam Stone, John Prine’s other saddest song would be *Hello in There, about aging:
You know that old trees just grow stronger,
And old rivers grow wilder every day,
Old people just grow lonesome,
Waiting for someone to say,
Hello in there.
Then there’s Angel from Montgomery:
*I am an old woman,
Named after my mother,
My old man was another,
Child who’s grown old.
If dreams were lightning,
Thunder was desire,
This old house would have burned down,
A long time ago.*
Lots of sad songs by John Prine, and at least half of them have nothing to do with romance.
“Aqualung” by Jethro Tull is a pretty depressing song, describing a pervert who lives on the streets, has the hots for little girls, and apparently had TB. Sample lyrics:
“Do you still remember December’s foggy freeze?
When the ice that clings onto your beard is screaming agony.
And you snatch a rattling last breath with deep sea diver’s sound
And the flowers bloom like madness in the spring.”
OK, they aren’t exactly genius lyrics, but the whole concept is depressing.
“Gross Chapel-British Grenadiers” by The Fall is another sad/spooky one.
–
And I find it kind of funny
I find it kind of sad
The dreams in which I’m dying
Are the best I’ve ever had
Gary Jules - Mad World (Tears for Fears cover)
Pink Floyd and Velevet Underground would have many depressing, non-love related songs, “Wish You Were Here” as mentioned before particularly stands out.
Led Zeppelin actually has a few that might fit, like “That’s the Way”
“The Who by Numbers” is quite dark and very good.
I’ve had no luck finding any other music from this obscure band (if anyone has info on them, please let me know), but “Harvest Moon” by a group called Bedlam on the Reservoir Dogs soundtrack is good.
Routine day by Klaatu (“the life I live would even make a dead man yawn”).
Fischer Z had a couple, including The worker and Red skies over paradise.
Winter by Tori Amos.
Streets of London by Ralph McTell
Working Class Hero by John Lennon
I mention them in this order because fella bilong missus flodnak played them in that order one dismal rainy autumn evening. I asked him if he was trying to make me want to slit my wrists.
“End of the Rainbow” Richard and Linda Thompson